Long-Range Coherence and Multiple Steady States in a Lossy Qubit Array

We show that a simple experimental setting of a locally pumped and lossy array of two-level quantum systems can stabilize states with strong long-range coherence. Indeed, by explicit analytic construction, we show there is an extensive set of steady-state density operators, from minimally to maximally entangled, despite this being an interacting open many-body problem. Such nonequilibrium steady states arise from a hidden symmetry that stabilizes Bell pairs over arbitrarily long distances, with unique experimental signatures. We demonstrate a protocol by which one can selectively prepare these states using dissipation. Our findings are accessible in present-day experiments.

Figure 1

(a) Schematic setup showing coherent tunneling $J$ and incoherent pump and loss ${\gamma }_{\pm }$ at the center. The system has a strong dynamical symmetry that stabilizes long-range entangled particle-hole pairs at reflection-symmetric sites. (b) Single-particle density matrix $⟨{\widehat{b}}_i^{†}{\widehat{b}}_j⟩$ in the steady state with maximum number of Bell pairs. The center has occupation ${\gamma }_{+}/({\gamma }_{+}+{\gamma }_{-})$. (c) End-to-end coherence for different symmetry sectors $\eta$.

Figure 2

Log negativity $E_{\mathcal{N}}$, measuring entanglement between left and right halves, in steady states corresponding to different symmetry sectors $\eta$ and number of sites $L$, with ${\gamma }_{+}={\gamma }_{-}$.

Figure 3

End-to-end correlation during a selective preparation of different symmetry sectors $\eta$ for $L=7$. The shaded region shows the generation of Bell pairs from symmetric Fock states, ${\widehat{b}}_0^{†}{\prod }_{k=1}^{\eta }{\widehat{b}}_k^{†}{\widehat{b}}_{-k}^{†}|0⟩$, driven by the loss of JW fermions at the center with rate ${\gamma }_F=3J$. The white region shows subsequent dynamics with boson pump and loss rates ${\gamma }_{\pm }=2J$.